#7670231

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	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6632 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)(9,10,13,14)(11,12,15,16)(18,21)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16)(17,18,20,21)(19,22,23,24);(2,5)(6,8)(10,14)(11,15)(17,19,20,23)(18,22,21,24);(9,11)(10,12)(13,15)(14,16);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,15)(14,16);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,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	#72342134, #72342212, #72342223
maximal normal subgroup of	#72342134, #72342212, #72342223

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7577427);