#778669

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3432 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)(5,6,8,11)(7,10,12,9)(13,14,17,20)(15,16,19,18)(21,22,26,28)(23,24,25,27);(5,7,8,12)(6,9,11,10)(13,14)(15,16)(17,20)(18,19)(21,23,26,25)(22,27,28,24);(13,15)(14,18)(16,20)(17,19)(21,24)(22,23)(25,28)(26,27);(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,25,26,23)(22,24,28,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,19)(15,20)(17,18)(21,24)(22,25)(23,28)(26,27)

Polycyclic presentation:

2^9|a^2=f,b^2=fh,d^2=gh,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.

Relationships

maximal subgroup of	#72331963, #72332005, #72332031
maximal normal subgroup of	#72331963, #72332005, #72332031
maximal quotient of	#72332557

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 685865);