#7503857

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	C₂×#3035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2202 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)(13,14,18,23,19,24,20,25)(15,17,22,28,27,21,26,16);(1,2)(3,4)(5,7,9,11)(6,10,12,8)(13,15)(14,21)(16,23)(17,24)(18,22)(19,27)(20,26)(25,28);(2,4)(5,8,9,10)(6,11,12,7)(13,16,19,28)(14,22,24,26)(15,25,27,23)(17,18,21,20);(6,12)(7,11)(13,17)(14,15)(16,18)(19,21)(20,28)(22,25)(23,26)(24,27);(13,18,19,20)(14,23,24,25)(15,26,27,22)(16,21,28,17)

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7411053);