#244512

order	512
nilpotency class	3
isoclinism	isoclinic class of #244499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1133 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1133 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151708);