#7156324

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	C₂×#3271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2038 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)(7,11)(10,12)(13,14,20,36,19,24,18,25)(15,26,35,21,28,41,17,23)(16,29,37,40,32,22,33,44)(27,43,31,39,30,42,34,38);(6,9)(7,11)(13,15)(14,21)(16,30)(17,20)(18,35)(19,28)(22,39)(23,24)(25,41)(26,36)(27,32)(29,38)(31,37)(33,34)(40,42)(43,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,32)(14,22,24,29)(15,27,28,30)(17,34,35,31)(18,33,20,37)(21,38,23,39)(25,40,36,44)(26,42,41,43);(6,9)(10,12)(13,17,20,15,19,35,18,28)(14,23,36,26,24,21,25,41)(16,31,37,30,32,34,33,27)(22,38,44,43,29,39,40,42);(5,7,8,11)(6,10,9,12)(14,24)(15,28)(16,32)(17,35)(25,36)(33,37)(38,39)(42,43)

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=fi,e^2=hi,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, 7063520);