#238735

order	512
nilpotency class	3
isoclinism	isoclinic class of #233984
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1083 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 145931);