#6379176

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

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	C₂²×#575	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4652 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)(3,4)(5,6,10,14,11,15,9,23)(7,16,19,12,18,28,24,27)(8,13,21,31,22,32,25,30)(17,33,35,26,34,29,20,36);(5,7)(6,12)(8,20)(9,24)(10,19)(11,18)(13,29)(14,28)(15,27)(16,23)(17,21)(22,35)(25,34)(26,30)(31,36)(32,33);(2,4);(5,8)(6,13)(7,17)(9,25)(10,21)(11,22)(12,26)(14,31)(15,32)(16,33)(18,34)(19,35)(20,24)(23,30)(27,36)(28,29);(7,18)(12,27)(16,28)(17,34)(19,24)(20,35)(26,36)(29,33)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2955
maximal normal subgroup of	D₄×#2955
maximal quotient of	#72286268, #72322064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286372);