#493361

order	512
nilpotency class	4
isoclinism	isoclinic class of #480250
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×#3449	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3438 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)(7,17)(8,13)(9,14)(10,24,25,15)(11,16)(12,19,27,34)(18,31)(20,32)(21,30,23,28)(22,35)(26,33,29,36);(5,7)(6,12)(8,18)(9,22)(10,19)(11,20)(13,26)(14,29)(15,17)(16,27)(21,33)(23,36)(24,32)(25,34)(28,31)(30,35);(2,4)(5,8,11,9)(6,13,16,14)(7,18,20,22)(10,21,25,23)(12,26,27,29)(15,28,24,30)(17,31,32,35)(19,33,34,36);(5,9,11,8)(6,14,16,13)(7,18,20,22)(10,23,25,21)(12,26,27,29)(15,30,24,28)(17,31,32,35)(19,33,34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240225, #72256596, #72261199, …
maximal normal subgroup of	#72240225, #72256596, #72261199, …
maximal quotient of	#72236572

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400557);