#7631188

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

8382 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)(17,25)(21,26)(23,28)(24,27);(1,2)(3,4)(13,14)(15,20)(16,22)(17,21)(18,19)(23,27)(24,28)(25,26);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(15,22)(21,26)(23,28);(13,15,18,22)(14,20,19,16)(17,23,25,28)(21,27,26,24);(13,16,18,20)(14,15,19,22)(17,24,25,27)(21,23,26,28);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)

Polycyclic presentation:

2^9|d^2=i,e^2=i,f^2=gh,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	#72304554, #72304599, #72304616
maximal normal subgroup of	#72304554, #72304599, #72304616

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538384);