#297316

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204512);