#376945

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,11,15)(7,17,19,28)(8,12,22,29)(9,23,26,31)(10,16,20,14)(13,25,30,24)(18,27,34,36)(21,35,33,32);(5,7)(6,12)(8,20)(9,18)(10,22)(11,19)(13,27)(14,28)(15,29)(16,17)(21,24)(23,32)(25,33)(26,34)(30,36)(31,35);(5,8,10,7,11,22,20,19)(6,12,14,28,15,29,16,17)(9,21,25,18,26,33,24,34)(13,27,31,35,30,36,23,32);(1,3,4,2)(5,9)(6,13)(7,18)(8,21)(10,25)(11,26)(12,27)(14,31)(15,30)(16,23)(17,32)(19,34)(20,24)(22,33)(28,35)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286523, #72286995, #72287028
maximal normal subgroup of	#72286523, #72286995, #72287028

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284141);