#373868

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	#2408
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1282 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)(7,19)(8,23)(9,27)(10,17)(11,16)(12,18)(13,21)(14,25)(15,24)(20,31)(22,36)(26,32)(28,35)(29,33)(30,34);(5,7,10,21,12,22,11,29)(6,13,16,19,18,33,17,36)(8,24,25,28,26,20,30,9)(14,31,23,35,34,15,32,27);(1,3,4,2)(5,8)(6,14)(7,20)(9,21)(10,25)(11,30)(12,26)(13,31)(15,33)(16,23)(17,32)(18,34)(19,35)(22,24)(27,36)(28,29);(5,9,10,24,12,28,11,20)(6,15,16,27,18,31,17,35)(7,8,21,25,22,26,29,30)(13,32,19,14,33,23,36,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281064);