#232427

order	512
nilpotency class	3
isoclinism	isoclinic class of #232415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1394 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1394 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,16)(15,18)(19,20)(21,22)(23,31)(24,30)(25,29)(26,28)(27,32)(33,35)(34,36);(8,15)(10,16)(11,18)(13,19)(21,23,26,34)(22,27,28,35)(24,33,25,32)(29,36,30,31);(2,4)(5,7)(6,10)(8,13)(9,14)(11,17)(12,16)(15,19)(18,20)(21,22)(23,31)(24,30)(25,29)(26,28)(27,32)(33,35)(34,36);(5,8,9,15)(6,11,12,18)(7,13,14,19)(10,17,16,20)(22,28)(27,35)(29,30)(31,36)

Polycyclic presentation:

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

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, 139623);