#6398876

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×#547	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2008 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)(15,20)(16,28)(17,22)(18,32)(19,23)(21,30)(24,40)(25,35)(26,36)(27,42)(29,44)(31,38)(33,39)(34,41)(37,43);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,29)(17,25)(18,27)(19,26)(21,37)(22,35)(23,36)(24,31)(28,44)(30,43)(32,42)(33,41)(34,39)(38,40);(13,16)(14,21)(15,24)(17,30)(18,33)(19,31)(20,34)(22,38)(23,39)(25,41)(26,29)(27,43)(28,32)(35,44)(36,37)(40,42);(7,12)(9,11)(13,17,19,18)(14,22,23,32)(15,25,26,27)(16,30,31,33)(20,35,36,42)(21,38,39,28)(24,41,29,43)(34,44,37,40);(1,3,4,2)(6,10)(9,11)(15,26)(20,36)(24,29)(25,27)(34,37)(35,42)(40,44)(41,43)

Polycyclic presentation:

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

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, 6306072);