#97450

order	512
nilpotency class	3
isoclinism	isoclinic class of #97444
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

1016 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 1016 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4646);