#392102

order	512
nilpotency class	3
isoclinism	isoclinic class of #32712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

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

862 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 282 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:

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

Polycyclic presentation:

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

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, 299298);