#6374177

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3489	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4412 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)(15,20)(16,28)(17,22)(18,33)(19,24)(21,30)(23,26)(25,38)(27,36)(29,43)(31,40)(32,37)(34,39)(35,42)(41,44);(5,7)(6,9)(8,12)(10,11)(13,15,19,27)(14,20,24,36)(16,25,31,41)(17,26,32,18)(21,35,39,43)(22,23,37,33)(28,38,40,44)(29,30,42,34);(7,12)(9,11)(13,16,19,31)(14,21,24,39)(15,25,27,41)(17,29,32,42)(18,34,26,30)(20,35,36,43)(22,38,37,44)(23,40,33,28);(6,10)(9,11)(13,17)(14,22)(15,26)(16,29)(18,27)(19,32)(20,23)(21,38)(24,37)(25,30)(28,43)(31,42)(33,36)(34,41)(35,40)(39,44);(15,27)(17,32)(20,36)(22,37)(25,41)(29,42)(35,43)(38,44)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281373);