#194480

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1096 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 258 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:

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

Polycyclic presentation:

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

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, 101676);