#6503002

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#37273	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2131 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^c=ei,e^d=ei

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, 6410198);