#7104330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996315
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25533	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=hi,d^2=f,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307698, #72307728, #72308361
maximal normal subgroup of	#72307698, #72307728, #72308361

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7011526);