#7006240

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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	#39956	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2057 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6913436);