#368597

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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

878 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275793);