#6399610

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×#548	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2169 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)(15,25)(16,30)(17,22)(18,23)(19,34)(20,28)(21,31)(24,33)(26,36)(27,40)(29,37)(32,39)(35,43)(38,42)(41,44);(6,9)(7,11)(13,15,18,28)(14,20,23,25)(16,27,17,26)(19,29,33,44)(21,38,39,35)(22,40,30,36)(24,37,34,41)(31,43,32,42);(13,16,18,17)(14,21,23,39)(15,26,28,27)(19,32,33,31)(20,35,25,38)(22,34,30,24)(29,43,44,42)(36,41,40,37);(5,7,8,11)(6,10,9,12)(13,17,18,16)(14,22,23,30)(15,27,28,26)(19,32,33,31)(20,36,25,40)(21,34,39,24)(29,43,44,42)(35,41,38,37);(1,3,4,2)(6,9)(10,12)(14,23)(16,17)(20,25)(24,34)(26,27)(31,32)(37,41)(42,43)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=f,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^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, 6306806);