#374100

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2508 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=fhi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,e^d=ei

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, 281296);