#823387

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2557 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=fgh,d^2=i,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 730583);