#7041292

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#44871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1591 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,5,4)(3,7,8,6)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,34)(28,31)(29,32)(30,38)(33,35)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,26,29,32)(27,30,37,39)(28,33,35,31)(34,36,40,38);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,35,36,29,37,28,38)(26,30,31,40,32,39,33,34);(3,8)(6,7)(12,20)(15,24)(17,19)(21,23)(25,26)(27,30)(28,33)(29,32)(31,35)(34,38)(36,40)(37,39);(10,16)(11,18)(15,24)(17,19)(25,28,29,35)(26,31,32,33)(27,36,37,38)(30,34,39,40)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=fg,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308949, #72308953
maximal normal subgroup of	#72308949, #72308953

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6948488);