#6375469

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34632	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,22,18,21,28,34)(15,19,25,31,24,30,37,41)(16,20,27,33,26,32,38,42)(23,29,35,40,36,39,43,44);(5,7)(6,9)(8,12)(10,11)(13,15,17,25,18,24,28,37)(14,19,22,31,21,30,34,41)(16,23,27,35,26,36,38,43)(20,29,33,40,32,39,42,44);(2,4)(7,12)(9,11)(13,16)(14,20)(15,23)(17,27)(18,26)(19,29)(21,32)(22,33)(24,36)(25,35)(28,38)(30,39)(31,40)(34,42)(37,43)(41,44);(6,10)(9,11)(15,24)(16,26)(19,30)(20,32)(25,37)(27,38)(31,41)(33,42);(14,21)(19,30)(20,32)(22,34)(29,39)(31,41)(33,42)(40,44)

Polycyclic presentation:

2^9|a^2=h,b^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,d^b=di,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216185
maximal normal subgroup of	#72216185
maximal quotient of	#72215932

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6282665);