#362719

order	512
nilpotency class	3
isoclinism	isoclinic class of #184811
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2610 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,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, 269915);