#364441

order	512
nilpotency class	3
isoclinism	isoclinic class of #196344
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#35088	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2586 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)(7,11)(8,18,21,12)(9,13)(10,14)(15,26,19,23)(16,24)(17,25)(20,31,33,27)(22,28)(29,36,32,34)(30,35);(2,4)(5,7)(6,11)(8,19,21,15)(9,16)(10,17)(12,26,18,23)(13,24)(14,25)(20,32,33,29)(22,30)(27,36,31,34)(28,35);(5,8)(6,12)(7,15)(9,20)(10,21)(11,23)(13,27)(14,18)(16,29)(17,19)(22,33)(24,34)(25,26)(28,31)(30,32)(35,36);(2,4)(5,9,10,22)(6,13,14,28)(7,16,17,30)(8,20,21,33)(11,24,25,35)(12,27,18,31)(15,29,19,32)(23,34,26,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265848, #72265850, #72314372, …
maximal normal subgroup of	#72265848, #72265850, #72314372, …
maximal quotient of	#72265993

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271637);