#464741

order	512
nilpotency class	3
isoclinism	isoclinic class of #454968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#23444	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1683 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)(9,21)(10,14)(13,22)(16,23,31,25)(17,24,32,29)(19,26)(20,27)(28,33)(30,34,36,35);(1,2,3,4)(5,7)(6,11)(8,16)(9,17)(10,18)(12,25)(13,24)(14,15)(19,30)(20,31)(21,29)(22,32)(23,27)(26,35)(28,34)(33,36);(2,4)(5,8)(6,12)(7,16)(9,19)(10,20)(11,23)(13,28)(14,27)(15,25)(17,30)(18,31)(21,26)(22,33)(24,35)(29,34)(32,36);(5,9,10,22)(6,13)(7,17,18,32)(8,19,20,33)(11,24)(12,26)(14,21)(15,29)(16,30,31,36)(23,34)(25,35)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251368, #72251399, #72251401, …
maximal normal subgroup of	#72251368, #72251399, #72251401, …
maximal quotient of	#72251470

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371937);