#348117

order	512
nilpotency class	3
isoclinism	isoclinic class of #348108
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

956 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 160 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,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, 255313);