#333379

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#1624	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

810 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=h,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240575);