#6395368

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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	#52169	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=efi,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, 6302564);