#230430

order	512
nilpotency class	3
isoclinism	isoclinic class of #20158
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2361
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,f^a=fi,d^b=de,e^b=eh,g^b=gh,d^c=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 137626);