#246314

order	512
nilpotency class	3
isoclinism	isoclinic class of #246239
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1143 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,e^b=eh,f^b=fi,e^c=ei,e^d=eh,f^d=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153510);