#8093661

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

6796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241092, #72241148, #72241152
maximal normal subgroup of	#72241092, #72241148, #72241152

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000857);