#7504200

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1865 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)(13,14,19,24,18,23,34,39)(15,25,28,40,27,20,35,37)(16,29,30,22,17,31,33,21)(26,41,42,36,32,44,43,38);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,20)(16,26)(17,32)(18,27)(19,28)(21,36)(22,38)(23,25)(24,37)(29,44)(30,42)(31,41)(33,43)(34,35)(39,40);(2,4)(13,16,18,17)(14,21,23,22)(15,26,27,32)(19,33,34,30)(20,36,25,38)(24,31,39,29)(28,43,35,42)(37,41,40,44);(6,11)(7,12)(13,17,18,16)(14,22,23,21)(15,26,27,32)(19,30,34,33)(20,36,25,38)(24,29,39,31)(28,43,35,42)(37,41,40,44);(5,8,9,10)(6,7,11,12)(14,23)(15,27)(21,22)(24,39)(26,32)(28,35)(29,31)(42,43)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,e^a=ei,c^b=ch,d^b=df,e^b=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, 7411396);