#6382411

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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

1979 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)(13,14,19,15,18,20,17,22)(16,21,25,23,26,28,27,24);(5,6)(7,10)(8,9)(11,12)(13,15,19,20,18,22,17,14)(16,24,25,21,26,23,27,28);(2,4)(14,20)(15,22)(21,28)(23,24);(6,9)(10,12)(13,16)(14,21)(15,23)(17,27)(18,26)(19,25)(20,28)(22,24);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,20,15)(16,25,26,27)(21,23,28,24)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289607);