#7629590

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

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

8187 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,f^2=g,b^a=bg,c^a=ch,c^b=ci,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536786);