#6382412

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2424	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289608);