#371186

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1107 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)(7,18)(8,14)(9,26)(10,29)(11,16)(12,17)(13,20)(15,27)(19,32)(21,36)(22,30)(23,33)(24,34)(25,31)(28,35);(1,3,4,2)(5,7,10,20,12,21,11,22)(6,13,16,18,17,30,29,36)(8,23,24,9,19,27,25,28)(14,26,31,33,32,35,34,15);(5,8,12,19)(6,14,17,32)(7,9,21,28)(10,25,11,24)(13,15,30,33)(16,34,29,31)(18,35,36,26)(20,23,22,27);(5,9)(6,15,17,33)(7,19)(8,21)(10,27)(11,23)(12,28)(13,14,30,32)(16,26,29,35)(18,31,36,34)(20,25)(22,24)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,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, 278382);