#7670629

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

5659 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=g,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304404, #72304407, #72304441
maximal normal subgroup of	#72304404, #72304407, #72304441

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577825);