#6371298

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#60387, #72312244, #72312515, …
maximal normal subgroup of	C₄×#60387, #72312244, #72312515, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278494);