#789600

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3906 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)(7,11)(8,12)(9,10)(14,18)(15,20)(21,22,25,24)(23,27,28,26);(6,10)(8,11)(13,14,17,18)(15,19,20,16)(21,22)(23,27)(24,25)(26,28);(1,2)(3,4)(21,23)(22,26)(24,27)(25,28);(5,7)(6,11)(8,10)(9,12)(13,15)(14,19)(16,18)(17,20)(21,23)(22,27)(24,26)(25,28);(5,8,9,11)(6,12,10,7)(13,16,17,19)(14,20,18,15)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=f,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, 696796);