#405612

order	512
nilpotency class	3
isoclinism	isoclinic class of #400057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319687, #72319923, #72320018
maximal normal subgroup of	#72319687, #72319923, #72320018

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312808);