#6481848

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470349
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2018 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314849, #72314856, #72314860, …
maximal normal subgroup of	#72314849, #72314856, #72314860, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389044);