#6407978

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2679	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,5)(4,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,20,23,21,24,27,30)(19,22,26,29,25,28,31,32);(1,3)(2,5)(4,7)(6,8)(9,11,12,16)(10,13,14,15);(2,6)(3,7)(11,16)(13,15)(19,25)(22,28)(26,31)(29,32);(2,6)(5,8)(10,14)(11,16);(17,19,20,26,21,25,27,31)(18,22,23,29,24,28,30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=cfg,d^b=df,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340637, #72340657, #72340675
maximal normal subgroup of	#72340637, #72340657, #72340675

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6315174);