#6403684

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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×#547	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 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)(7,11)(8,10)(9,12)(13,14)(15,19)(16,27)(17,21)(18,23)(20,29)(22,28)(24,37)(25,32)(26,33)(30,35)(31,39)(34,38)(36,41)(40,42)(43,44);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,24)(17,25)(18,26)(20,31)(21,32)(22,34)(23,33)(27,37)(28,38)(29,39)(30,40)(35,42)(36,43)(41,44);(6,10)(9,11)(13,16,28,41,18,30,17,29)(14,20,21,35,23,36,22,27)(15,24,38,44,26,40,25,39)(19,31,32,42,33,43,34,37);(7,12)(9,11)(13,17,18,28)(14,21,23,22)(15,25,26,38)(16,29,30,41)(19,32,33,34)(20,35,36,27)(24,39,40,44)(31,42,43,37);(1,3,4,2)(6,10)(9,11)(15,26)(16,30)(19,33)(20,36)(25,38)(27,35)(29,41)(32,34)

Polycyclic presentation:

2^9|a^2=h,c^2=fgi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=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, 6310880);