#417844

order	512
nilpotency class	3
isoclinism	isoclinic class of #417487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2811 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286994, #72287334, #72287352
maximal normal subgroup of	#72286994, #72287334, #72287352

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325040);