#6444376

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279716, #72279895, #72280641, …
maximal normal subgroup of	#72279716, #72279895, #72280641, …
maximal quotient of	#72313327

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351572);