#238649

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37639	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1136 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,d^c=dhi,f^c=fh,g^c=ghi

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, 145845);