#364503

order	512
nilpotency class	3
isoclinism	isoclinic class of #23872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2422 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)(5,6,10,14)(7,11,17,25)(8,12,18,26)(9,20,21,13)(15,28,22,23)(16,30,31,24)(19,33,32,27)(29,35,34,36);(1,3,4,2)(7,15)(9,21)(11,23)(13,20)(16,29)(17,22)(19,32)(24,35)(25,28)(27,33)(30,36)(31,34);(5,7)(6,11)(8,16)(9,22)(10,17)(12,24)(13,28)(14,25)(15,21)(18,31)(19,34)(20,23)(26,30)(27,36)(29,32)(33,35);(5,8)(6,12)(7,16)(9,19)(10,18)(11,24)(13,27)(14,26)(15,29)(17,31)(20,33)(21,32)(22,34)(23,35)(25,30)(28,36)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gi,e^c=ei,g^c=gi,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, 271699);