#7631003

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

8378 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)(10,14)(15,16)(19,23)(20,24);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,21,22)(19,24,23,20);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,19)(18,20)(21,23)(22,24);(2,5)(6,8)(11,12)(15,16)(17,18,21,22)(19,24,23,20);(1,2)(3,6)(4,5)(7,8)(9,12,13,11)(10,16,14,15)(17,18)(19,20)(21,22)(23,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,20)(18,19)(21,24)(22,23)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=h,e^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	#72342147, #72342197, #72342206
maximal normal subgroup of	#72342147, #72342197, #72342206

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538199);