#493608

order	512
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
socle	C₂²	C₂×B(4,𝔽₂)
prime radical	C₂×#3529	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2433 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2433 conjugacy classes of subgroups — too many to draw. Only its 314 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)(3,4)(5,6,11,14,12,17,10,16)(7,18,21,34,22,29,20,35)(8,15,24,32,9,25,23,33)(13,27,28,26,31,36,30,19);(5,7)(6,13)(8,19)(9,26)(10,21)(11,20)(12,22)(14,28)(15,29)(16,30)(17,31)(18,25)(23,27)(24,36)(32,35)(33,34);(2,4)(6,14)(10,11)(13,28)(15,32)(16,17)(18,34)(20,21)(23,24)(25,33)(27,36)(29,35)(30,31);(5,8,12,9)(6,15,17,25)(7,19,22,26)(10,23,11,24)(13,29,31,18)(14,32,16,33)(20,36,21,27)(28,35,30,34)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^e=fi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261524, #72261597, #72261670
maximal normal subgroup of	#72261524, #72261597, #72261670
maximal quotient of	#72261886

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400804);