#364613

order	512
nilpotency class	3
isoclinism	isoclinic class of #364611
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13967
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60367	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2387 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338602, #72338605, #72338612
maximal normal subgroup of	#72338602, #72338605, #72338612

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271809);