#6372192

order	512
nilpotency class	3
isoclinism	isoclinic class of #60367
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3542 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)(7,11)(8,9)(10,12)(13,14,19,21,18,15,17,22)(16,23,26,28,24,20,25,27);(1,2,3,4)(6,9)(10,11)(13,15,18,14)(16,23,24,20)(17,22,19,21)(25,28,26,27);(5,7,8,12)(6,10,9,11)(13,16,18,24)(14,20,15,23)(17,25,19,26)(21,27,22,28);(16,24)(20,23)(25,26)(27,28);(13,17,18,19)(14,21,15,22)(16,25,24,26)(20,27,23,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279388);