#7631329

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
prime radical	D₄×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6235 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233523, #72233541, #72233553, …
maximal normal subgroup of	#72233523, #72233541, #72233553, …
maximal quotient of	#72262040

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538525);