#387628

order	512
nilpotency class	3
isoclinism	isoclinic class of #232138
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1032 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1032 conjugacy classes of subgroups — too many to draw. Only its 242 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294824);