#7504207

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1842 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)(13,14,18,23,19,22,32,37)(15,20,27,34,26,35,42,44)(16,28,29,36,17,30,31,21)(24,38,39,43,25,40,41,33);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,26)(14,20,22,35)(16,24,17,25)(18,27,32,42)(21,33,36,43)(23,34,37,44)(28,38,30,40)(29,39,31,41);(2,4)(13,16,19,17)(14,21,22,36)(15,24,26,25)(18,31,32,29)(20,33,35,43)(23,30,37,28)(27,41,42,39)(34,40,44,38);(6,11)(7,12)(13,17,19,16)(14,21,22,36)(15,25,26,24)(18,29,32,31)(20,33,35,43)(23,30,37,28)(27,39,42,41)(34,40,44,38);(5,8,9,10)(6,7,11,12)(14,22)(15,26)(21,36)(23,37)(24,25)(27,42)(28,30)(39,41)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=f,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, 7411403);