#7633107

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

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

8128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 20 points:

(2,4)(13,14,17,16)(15,19,20,18);(1,2,3,4)(6,8)(10,12)(13,14,17,16)(15,18,20,19);(5,6,9,8)(7,10,11,12)(13,15,17,20)(14,18,16,19);(7,11)(10,12)(13,16,17,14)(15,19,20,18);(5,7)(6,10)(8,12)(9,11);(5,8,9,6)(7,12,11,10)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=h,f^2=i,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540303);