#458574

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16320
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 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)(3,7,14,23)(4,8)(5,18)(6,10)(9,17)(11,20)(12,27,13,21)(15,31,30,24)(16,25)(19,26)(22,28,32,29);(1,3)(2,7,10,23)(4,13)(5,15)(6,14)(8,21,25,27)(9,22,26,32)(11,24,18,31)(12,16)(17,29)(19,28)(20,30);(1,4)(2,8)(3,12,14,13)(5,19)(6,16)(7,21,23,27)(9,11)(10,25)(15,29,30,28)(17,20)(18,26)(22,31,32,24);(1,4)(2,9)(3,13)(5,19)(6,16)(7,22)(8,18)(10,26)(11,25)(12,14)(15,28)(17,20)(21,31)(23,32)(24,27)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319370, #72319383, #72319444
maximal normal subgroup of	#72319370, #72319383, #72319444

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 365770);