#412968

order	512
nilpotency class	3
isoclinism	isoclinic class of #396724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

988 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320164);