#412824

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

996 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^c=ei,f^c=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, 320020);