#195700

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1458 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1458 conjugacy classes of subgroups — too many to draw. Only its 274 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102896);