#501133

order	512
nilpotency class	4
isoclinism	isoclinic class of #501132
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1158 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 140 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fhi,e^c=eh,f^c=fi,e^d=ei,f^d=fhi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286789, #72286802
maximal normal subgroup of	#72286789, #72286802

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408329);