#362964

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1098 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=g,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270160);