#9888132

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

5672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256306, #72263026, #72263033
maximal normal subgroup of	#72256306, #72263026, #72263033

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9795328);