#7185480

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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	#59145	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1909 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092676);