#380555

order	512
nilpotency class	3
isoclinism	isoclinic class of #214664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1663 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285194, #72285387, #72285542
maximal normal subgroup of	#72285194, #72285387, #72285542

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287751);