#540048

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=hi,d^2=fgi,e^2=fgh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 447244);