#7630956

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	#34981	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7078 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)(12,15)(18,21)(22,24);(2,5)(6,8)(9,10,13,14)(11,15,16,12)(17,18)(19,22)(20,21)(23,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,22)(20,21)(23,24);(2,5)(6,8)(9,11,13,16)(10,15,14,12)(17,19,20,23)(18,22,21,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)

Polycyclic presentation:

2^9|b^2=gh,d^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	#72241084, #72241089, #72241090, …
maximal normal subgroup of	#72241084, #72241089, #72241090, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538152);