#8077599

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

5593 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256180, #72263026, #72263028
maximal normal subgroup of	#72256180, #72263026, #72263028

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7984795);