#371189

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

1109 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)(8,23)(9,27)(10,30)(11,16)(12,17)(13,20)(14,25)(15,28)(19,36)(21,33)(22,32)(24,35)(26,34)(29,31);(1,3,4,2)(5,7,10,20,12,22,11,21)(6,13,16,18,17,33,30,32)(8,24,25,9,19,28,26,29)(14,15,23,27,34,35,36,31);(5,8,12,19)(6,14,17,34)(7,9,22,29)(10,26,11,25)(13,31,33,27)(15,18,35,32)(16,36,30,23)(20,24,21,28);(5,9)(6,15,17,35)(7,19)(8,22)(10,28)(11,24)(12,29)(13,23,33,36)(14,18,34,32)(16,27,30,31)(20,26)(21,25)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=eg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,f^b=fi,e^c=ei,f^c=fi,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, 278385);