#7500911

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500908
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,f^b=fi,h^b=hi,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306459, #72306460, #72306464
maximal normal subgroup of	#72306459, #72306460, #72306464

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7408107);