#6399594

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306790);