#355101

order	512
nilpotency class	3
isoclinism	isoclinic class of #21776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1972 conjugacy classes of subgroups; only the 394 normal subgroups are shown.

normal subgroups only

This group has 1972 conjugacy classes of subgroups — too many to draw. Only its 394 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=ce,f^b=fi,g^b=gi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262297);