#7716016

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

7624 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)(5,6,10,8)(7,12,9,11)(13,14)(15,20)(16,18)(17,25)(19,21)(22,26)(23,27)(24,28);(1,2)(3,4)(5,7,10,9)(6,11,8,12)(13,15,19,16)(14,20,21,18)(17,24,26,23)(22,27,25,28);(6,8)(11,12)(14,21)(15,16)(17,26)(27,28);(5,8,10,6)(7,12,9,11)(13,16,19,15)(14,18,21,20)(17,24,26,23)(22,27,25,28);(5,9,10,7)(6,12,8,11)(13,17,19,26)(14,22,21,25)(15,23,16,24)(18,27,20,28);(13,18)(14,15)(16,21)(17,27)(19,20)(22,23)(24,25)(26,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7623212);