#235528

order	512
nilpotency class	3
isoclinism	isoclinic class of #235516
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#37606	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1301 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,d^2=hi,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286549, #72286719, #72286735, …
maximal normal subgroup of	#72286549, #72286719, #72286735, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142724);