#516580

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

6742 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(6,8)(9,10,11,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,23)(22,27)(26,28);(2,4)(6,8)(9,10,11,12)(14,18)(19,21)(20,24)(26,27);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28);(13,16,17,25)(14,20,18,24)(15,22,23,28)(19,26,21,27)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315658, #72315672, #72315688, …
maximal normal subgroup of	#72315658, #72315672, #72315688, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 423776);