#392516

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

958 conjugacy classes of subgroups; only the 354 normal subgroups are shown.

normal subgroups only

This group has 958 conjugacy classes of subgroups — too many to draw. Only its 354 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=fi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299712);