#6465653

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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

3430 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)(9,10,13,17,14,18,15,12)(11,19,22,20,21,16,23,24);(2,4)(9,10)(11,20)(12,13)(14,18)(15,17)(16,23)(19,22)(21,24);(5,6,7,8)(9,11,13,22,14,21,15,23)(10,16,17,24,18,19,12,20);(9,12,13,10,14,17,15,18)(11,20,22,16,21,24,23,19);(11,21)(16,19)(20,24)(22,23)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312807, #72312866, #72312956
maximal normal subgroup of	#72312807, #72312866, #72312956
maximal quotient of	#72315862

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372849);