#7502790

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#30704	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7409986);