#115136

order	512
nilpotency class	3
isoclinism	isoclinic class of #100203
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂⁵	C₂²×C₄
Frattini, prime radical, cosocle	C₂⁶	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

1638 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1638 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,g^b=gh,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22332);