#494998

order	512
nilpotency class	4
isoclinism	isoclinic class of #33688
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 ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

568 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)(5,6)(7,17)(8,14)(9,24)(10,26)(11,16)(12,18)(13,23)(15,22)(19,28)(20,35)(21,31)(25,33)(27,32)(29,34)(30,36);(1,3,4,2)(8,21)(9,23)(13,24)(14,31)(17,28)(18,34)(25,36)(32,35);(5,7)(6,12)(8,18)(9,25)(10,27)(11,19)(13,28)(14,32)(15,33)(16,29)(17,24)(20,22)(21,34)(23,36)(26,30)(31,35);(5,8,22,9,11,21,10,23)(6,13,26,31,16,24,15,14)(7,18,20,25,19,34,27,36)(12,28,30,35,29,17,33,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=ehi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,h^a=hi,d^b=di,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, 402194);