#353757

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

958 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240725, #72240734, #72240846, …
maximal normal subgroup of	#72240725, #72240734, #72240846, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260953);