#7626874

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gh,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534070);