#6437780

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35251	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4725 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)(9,10,13,16,14,15,22,23)(11,17,18,20,19,24,21,12);(5,6)(7,8)(9,11,14,19)(10,12,15,20)(13,21,22,18)(16,24,23,17);(2,4)(6,8)(9,12)(10,11)(13,17)(14,20)(15,19)(16,18)(21,23)(22,24);(6,8)(9,12)(10,11)(13,17)(14,20)(15,19)(16,18)(21,23)(22,24);(10,15)(12,20)(16,23)(17,24)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315616, #72315640, #72316096
maximal normal subgroup of	#72315616, #72315640, #72316096

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344976);