#6383703

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(2,6)(7,8)(10,13)(11,15)(17,18,22,26,21,25,30,32)(19,27,23,31,28,20,24,29);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19,22,23,21,28,30,24)(18,20,26,29,25,27,32,31);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20,21,27)(18,23,25,24)(19,26,28,32)(22,29,30,31);(2,6)(7,8)(10,13)(14,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,25)(28,32)(30,31);(1,4,5,3)(2,8,6,7)(19,28)(20,27)(23,24)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^b=dfi,e^b=ei,d^c=di,e^c=ei,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, 6290899);