#6399574

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

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×#3269	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^c=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, 6306770);