#358953

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

778 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)(7,9)(8,21)(10,19)(11,16)(12,17)(13,20)(14,25)(15,28)(18,31)(22,35)(23,32)(24,33)(26,34)(27,29)(30,36);(5,7)(6,13)(8,22,25,36)(9,19)(10,29)(11,20)(12,15)(14,31,21,23)(16,28)(17,27)(18,26,32,24)(30,33,35,34);(5,8,10,24,12,25,11,26)(6,14,16,33,17,21,19,34)(7,18,20,22,15,32,29,36)(9,23,27,35,28,31,13,30);(5,9,11,13,12,28,10,27)(6,15,16,29,17,7,19,20)(8,23,26,30,25,31,24,35)(14,32,33,36,21,18,34,22)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266149);