#417855

order	512
nilpotency class	3
isoclinism	isoclinic class of #417601
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

2817 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)(7,18)(8,14)(9,16)(10,15)(11,17)(12,21)(13,23)(19,27)(20,34)(22,33)(24,32)(25,30)(26,29)(28,36)(31,35);(5,7,9,21,11,22,10,26)(6,12,15,18,17,29,16,33)(8,19,23,35,24,36,25,20)(13,28,14,31,30,27,32,34);(5,8)(6,13)(7,19)(9,23)(10,25)(11,24)(12,27)(14,15)(16,32)(17,30)(18,34)(20,26)(21,35)(22,36)(28,29)(31,33);(2,4)(5,8)(6,14)(7,20)(9,25)(10,23)(11,24)(12,28)(13,15)(16,30)(17,32)(18,34)(19,26)(21,36)(22,35)(27,29)(31,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252185, #72256282, #72256291
maximal normal subgroup of	#72252185, #72256282, #72256291

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325051);