#147473

order	512
nilpotency class	4
isoclinism	isoclinic class of #13005
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1874
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10931
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dhi,h^b=hi,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307178, #72307205, #72307275, …
maximal normal subgroup of	#72307178, #72307205, #72307275, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54669);