#231702

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

642 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 198 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=fhi,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138898);