#6465643

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3374 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)(9,10,15,16,14,18,13,12)(11,17,22,23,21,24,19,20);(2,4)(10,16)(11,19)(12,18)(13,15)(20,23)(21,22);(5,6,7,8)(9,11,13,19,14,21,15,22)(10,17,12,20,18,24,16,23);(9,12,13,18,14,16,15,10)(11,20,19,24,21,23,22,17);(11,21)(17,24)(19,22)(20,23)

Polycyclic presentation:

2^9|a^2=fi,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312807, #72312851, #72312949
maximal normal subgroup of	#72312807, #72312851, #72312949
maximal quotient of	#72315676

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372839);