#7504109

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#35156	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411305);