#6389615

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34547	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22,27,23,26,29,25,30)(24,28,34,31,33,36,35,32);(5,7)(6,11)(8,15)(9,17)(10,16)(12,19)(13,14)(18,20)(21,23,27,29,26,30,25,22)(24,32,34,28,33,31,35,36);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24,26,33)(22,28,29,36)(23,31,30,32)(25,35,27,34);(7,16)(11,14)(15,17)(19,20)(22,29)(23,30)(24,33)(34,35);(5,9,10,8)(6,12,13,18)(7,17,16,15)(11,19,14,20)(21,25,26,27)(22,30,29,23)(24,34,33,35)(28,31,36,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df,d^c=di,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280361, #72280436, #72282542
maximal normal subgroup of	#72280361, #72280436, #72282542

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6296811);