#946639

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3151 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)(5,6,9,12)(7,11,8,10)(13,14,17,19)(15,16,20,18)(22,25)(23,27);(5,7,9,8)(6,10,12,11)(13,15,17,20)(14,18,19,16)(21,22,24,25)(23,26,27,28);(7,8)(10,11)(14,19)(15,20)(21,22,24,25)(23,26,27,28);(1,2,3,4)(5,8,9,7)(6,11,12,10)(13,14)(15,16)(17,19)(18,20)(21,22)(23,26)(24,25)(27,28);(5,6)(7,10)(8,11)(9,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fh,d^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331961, #72331994, #72332037
maximal normal subgroup of	#72331961, #72331994, #72332037
maximal quotient of	#72332614

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 853835);