#6465661

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3388 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=gh,d^2=h,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	#72307380, #72307390, #72307511, …
maximal normal subgroup of	#72307380, #72307390, #72307511, …
maximal quotient of	#72315688

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372857);