#348132

order	512
nilpotency class	3
isoclinism	isoclinic class of #348128
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3459	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

930 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,c^b=ch,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255328);