#373638

order	512
nilpotency class	3
isoclinism	isoclinic class of #23895
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2378 conjugacy classes of subgroups; only the 346 normal subgroups are shown.

normal subgroups only

This group has 2378 conjugacy classes of subgroups — too many to draw. Only its 346 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:

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

Polycyclic presentation:

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

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, 280834);