#7199146

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1795 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=fh,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7106342);