#7195665

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2404 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)(13,14,18,23,19,24,33,39)(15,25,27,37,28,42,30,20)(16,29,32,44,31,21,38,36)(17,22,35,41,34,40,26,43);(6,9)(7,11)(13,15,19,28)(14,20,24,37)(16,26,31,35)(17,32,34,38)(18,30,33,27)(21,22,29,40)(23,42,39,25)(36,43,44,41);(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,16,19,31)(14,21,24,29)(15,26,28,35)(17,30,34,27)(18,32,33,38)(20,22,37,40)(23,36,39,44)(25,41,42,43);(6,9)(10,12)(13,17,33,26,19,34,18,35)(14,22,39,43,24,40,23,41)(15,16,30,38,28,31,27,32)(20,36,42,21,37,44,25,29);(5,7,8,11)(6,10,9,12)(16,31)(17,34)(21,29)(22,40)(26,35)(32,38)(36,44)(41,43)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=fi,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, 7102861);