#6473098

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469983
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×#3191	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2167 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6380294);