#365371

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1494 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)(3,4)(5,6,10,14,11,15,9,23)(7,12,18,28,17,26,24,27)(8,13,22,31,20,30,21,35)(16,32,19,29,33,25,34,36);(2,4)(5,7,10,18,11,17,9,24)(6,12,14,28,15,26,23,27)(8,19,22,33,20,34,21,16)(13,29,31,25,30,36,35,32);(5,8)(6,13)(7,16,17,33)(9,21)(10,22)(11,20)(12,25,26,32)(14,31)(15,30)(18,19,24,34)(23,35)(27,29,28,36);(2,4)(8,20)(13,30)(16,33)(19,34)(21,22)(25,32)(29,36)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286985, #72286995, #72287084
maximal normal subgroup of	#72286985, #72286995, #72287084

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272567);