#7682881

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

7341 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590077);