#6393766

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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×#3561	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3116 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=df,e^b=ei,g^c=gi,h^c=hi

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, 6300962);