#400684

order	512
nilpotency class	3
isoclinism	isoclinic class of #395181
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1431 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=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, 307880);