#4327960

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4810 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232342, #72232357, #72232360, …
maximal normal subgroup of	#72232342, #72232357, #72232360, …
maximal quotient of	#72328438, C₂×#4327960

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4235156);