#364623

order	512
nilpotency class	3
isoclinism	isoclinic class of #23892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

2409 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=i,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,e^b=ei,g^b=gi,f^c=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, 271819);