#7504091

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	#31063	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1947 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7411287);