#6383862

order	512
nilpotency class	3
isoclinism	isoclinic class of #61049
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	D₄×#135	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,d^b=df,f^b=fi,e^c=ei,f^d=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2957
maximal normal subgroup of	D₄×#2957

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291058);