#427823

order	512
nilpotency class	3
isoclinism	isoclinic class of #427822
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15375
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1207 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12)(3,8,15,30)(4,9,20,35)(5,10,22,37)(6,24,27,11)(13,41,43,28)(14,36,21,29)(16,39,25,31)(17,32,45,58)(18,47,49,33)(19,40,26,34)(23,54,53,38)(42,62,51,56)(44,57,52,63)(46,59,48,60)(50,61,55,64);(1,3)(2,8)(4,17)(5,21)(6,25)(7,15)(9,32)(10,36)(11,39)(12,30)(13,19)(14,22)(16,27)(18,48)(20,45)(23,44)(24,31)(26,43)(28,34)(29,37)(33,60)(35,58)(38,57)(40,41)(42,55)(46,49)(47,59)(50,51)(52,53)(54,63)(56,64)(61,62);(1,4)(2,9)(3,13)(5,18)(6,26)(7,20)(8,28)(10,33)(11,40)(12,35)(14,42)(15,43)(16,45)(17,25)(19,27)(21,51)(22,49)(23,55)(24,34)(29,56)(30,41)(31,58)(32,39)(36,62)(37,47)(38,64)(44,48)(46,52)(50,53)(54,61)(57,60)(59,63);(1,5)(2,10)(3,14,15,21)(4,18)(6,23)(7,22)(8,29,30,36)(9,33)(11,38)(12,37)(13,42,43,51)(16,44,25,52)(17,46,45,48)(19,50)(20,49)(24,54)(26,55)(27,53)(28,56,41,62)(31,57,39,63)(32,59,58,60)(34,61)(35,47)(40,64)

Polycyclic presentation:

2^9|a^2=i,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286106, #72286134, #72286231
maximal normal subgroup of	#72286106, #72286134, #72286231

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335019);