#453812

order	512
nilpotency class	3
isoclinism	isoclinic class of #425083
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2576
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16125
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

701 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339028, #72339032, #72339042
maximal normal subgroup of	#72339028, #72339032, #72339042

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361008);