#227602

order	512
nilpotency class	3
isoclinism	isoclinic class of #20022
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	#2358
prime radical	#18494	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

621 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287666, #72322056, #72322149
maximal normal subgroup of	#72287666, #72322056, #72322149

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 134798);