#492990

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂²×D₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1310 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312332, #72312334, #72312349, …
maximal normal subgroup of	#72312332, #72312334, #72312349, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400186);