#371124

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

916 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)(5,6,12,18,10,16,11,17)(7,19,9,28,21,13,22,15)(8,23,26,31,27,14,25,32)(20,33,35,30,34,29,24,36);(1,3,4,2)(5,7,12,9,10,21,11,22)(6,13,18,15,16,19,17,28)(8,24,26,20,27,35,25,34)(14,30,32,29,23,36,31,33);(5,8)(6,14)(7,20)(9,24)(10,27)(11,26)(12,25)(13,29)(15,30)(16,23)(17,32)(18,31)(19,33)(21,34)(22,35)(28,36);(5,9,12,21,10,22,11,7)(6,15,18,19,16,28,17,13)(8,24,25,34,27,35,26,20)(14,30,31,33,23,36,32,29)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=eg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^b=fi,e^c=ei,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278320);