#191225

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,11)(4,7)(5,15)(6,10)(8,13)(9,12)(14,16)(17,18)(19,27,31,23)(20,32,34,47)(21,25)(22,26)(24,33,38,42)(28,35)(29,43)(30,40,41,37)(36,44)(39,48)(45,46);(1,3,12,14,6,13,5,4)(2,7,15,8,10,16,9,11)(17,19,28,41,22,31,21,30)(18,23,35,40,26,27,25,37)(20,29,42,48,34,45,33,44)(24,36,47,43,38,39,32,46);(1,4,5,13,6,14,12,3)(2,8,9,7,10,11,15,16)(17,20)(18,24,26,38)(19,29)(21,33)(22,34)(23,36,27,39)(25,32,35,47)(28,42)(30,44)(31,45)(37,46,40,43)(41,48);(17,21,22,28)(18,25,26,35)(19,30,31,41)(20,33,34,42)(23,37,27,40)(24,32,38,47)(29,44,45,48)(36,46,39,43)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=e,d^2=hi,e^2=i,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=ghi,g^c=ghi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98421);