#371027

order	512
nilpotency class	3
isoclinism	isoclinic class of #23998
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1320 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)(7,11,18,27)(8,13,20,30)(9,15)(10,14)(12,21,29,22)(16,25)(17,32)(19,26,34,35)(23,28)(24,33)(31,36);(1,3,4,2)(5,7)(6,11,14,27)(8,17)(9,19)(10,18)(12,24,29,36)(13,25,30,32)(15,26,28,35)(16,20)(21,33)(22,31)(23,34);(5,8)(6,12)(7,16)(9,22)(10,20)(11,24)(13,28)(14,29)(15,30)(17,18)(19,33)(21,23)(25,35)(26,32)(27,36)(31,34);(5,8)(6,13,14,30)(7,17)(9,22)(10,20)(11,25,27,32)(12,28,29,15)(16,18)(19,31)(21,23)(24,35,36,26)(33,34)

Polycyclic presentation:

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

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, 278223);