#6400312

order	512
nilpotency class	2
isoclinism	isoclinic class of #46350
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	#52129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1923 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)(5,6,9,12)(7,11,8,10)(13,14,20,23,19,24,17,28)(15,25,26,18,27,21,16,22);(5,6)(7,10)(8,11)(9,12)(13,15)(14,21)(16,17)(18,28)(19,27)(20,26)(22,23)(24,25);(2,4)(5,7,9,8)(6,10,12,11)(13,16,19,26)(14,22,24,18)(15,17,27,20)(21,23,25,28);(5,7,9,8)(6,11,12,10)(13,17,19,20)(14,23,24,28)(15,26,27,16)(18,25,22,21);(5,8,9,7)(6,11,12,10)(13,18,20,21,19,22,17,25)(14,15,23,26,24,27,28,16)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6307508);