#7502041

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1841 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6,9,12)(7,8,11,10)(14,20)(15,23)(16,25)(17,19)(21,24)(26,27);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,14)(15,16)(17,28)(18,20)(19,22)(21,27)(23,26)(24,25);(2,4)(5,8,9,10)(6,11,12,7)(13,15,18,21)(14,16,20,27)(17,23,19,24)(22,25,28,26);(6,12)(7,11)(13,16)(14,21)(15,20)(17,26)(18,27)(19,25)(22,23)(24,28);(13,17,18,19)(14,22,20,28)(15,24,21,23)(16,26,27,25)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7409237);