#7631130

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

8482 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304549, #72304601, #72304616
maximal normal subgroup of	#72304549, #72304601, #72304616

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538326);