#471162

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16768
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

630 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)(4,8,16,24)(5,10)(6,11)(9,17,25,27)(12,26,13,20)(14,23)(15,22)(18,19)(21,29,32,28)(30,31);(1,3,6,14,5,15,18,30)(2,7,19,31,10,22,11,23)(4,13,17,29,16,12,27,28)(8,20,9,21,24,26,25,32);(1,4,5,16)(2,8,10,24)(3,12)(6,17,18,27)(7,20)(9,11,25,19)(13,15)(14,28)(21,31)(22,26)(23,32)(29,30);(1,4)(2,9,10,25)(3,13)(5,16)(6,17)(7,21,22,32)(8,19,24,11)(12,15)(14,29)(18,27)(20,31,26,23)(28,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378358);