#6385051

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	#52160	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1985 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)(6,10)(11,12)(13,14,20,22,19,24,17,26)(15,18,25,28,16,21,23,27);(5,6,9,10)(7,11,8,12)(13,15,19,16)(14,21,24,18)(17,23,20,25)(22,27,26,28);(2,4)(5,7,9,8)(6,11,10,12)(13,16,19,15)(14,21,24,18)(17,25,20,23)(22,27,26,28);(5,8,9,7)(6,11,10,12)(13,17,19,20)(14,22,24,26)(15,25,16,23)(18,27,21,28);(13,18,20,28,19,21,17,27)(14,23,22,15,24,25,26,16)

Polycyclic presentation:

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

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, 6292247);