#492874

order	512
nilpotency class	4
isoclinism	isoclinic class of #33388
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1606 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)(5,6,10,15)(7,9,18,24)(8,13,19,27)(11,14,25,30)(12,26,23,16)(17,20,31,32)(21,28,33,36)(22,35,34,29);(5,7,11,16,12,14,24,15,10,18,25,26,23,30,9,6)(8,17,21,29,22,28,32,27,19,31,33,35,34,36,20,13);(5,8)(6,13)(7,17)(9,20)(10,19)(11,21)(12,22)(14,28)(15,27)(16,29)(18,31)(23,34)(24,32)(25,33)(26,35)(30,36);(8,19)(13,27)(17,31)(20,32)(21,33)(22,34)(28,36)(29,35)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226793, #72226794, #72226798, …
maximal normal subgroup of	#72226793, #72226794, #72226798, …
maximal quotient of	D₄×#1386

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400070);