#493428

order	512
nilpotency class	4
isoclinism	isoclinic class of #480268
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3485	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3446 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,g^b=gh,d^c=di,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240225, #72256597, #72261605, …
maximal normal subgroup of	#72240225, #72256597, #72261605, …
maximal quotient of	#72236774

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400624);