#7503809

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×#3041	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,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, 7411005);