#492928

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224839, #72224874, #72224928, …
maximal normal subgroup of	#72224839, #72224874, #72224928, …
maximal quotient of	#72224829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400124);