#494995

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322481
maximal normal subgroup of	#72322481

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402191);