#7631199

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

8186 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)(18,26)(22,24)(23,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14,19,16)(15,21,17,20)(18,22,26,24)(23,28,25,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,17)(14,20,16,21)(18,23,26,25)(22,27,24,28);(13,16,19,14)(15,20,17,21)(18,24,26,22)(23,27,25,28);(13,17,19,15)(14,21,16,20)(18,25,26,23)(22,28,24,27);(5,7,8,11)(6,10,9,12)(13,18)(14,22)(15,23)(16,24)(17,25)(19,26)(20,27)(21,28)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=i,f^2=g,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	#72304566, #72304611, #72304616
maximal normal subgroup of	#72304566, #72304611, #72304616

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538395);