#6445630

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2172 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)(7,11)(8,9)(10,12)(13,14,16,21,17,20,25,26)(15,19,23,27,22,28,24,18);(1,2)(3,4)(6,9)(10,11)(13,15,17,22)(14,18,20,27)(16,24,25,23)(19,21,28,26);(2,4)(5,7,8,12)(6,10,9,11)(13,15)(14,19)(16,23)(17,22)(18,26)(20,28)(21,27)(24,25);(2,4)(15,22)(18,27)(19,28)(23,24);(14,20)(18,27)(19,28)(21,26)

Polycyclic presentation:

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

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, 6352826);