#991593

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
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	C₂×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4312 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)(5,6,9,10)(7,12,11,8)(13,14)(15,17)(16,19)(18,20)(21,22,25,24)(23,27,28,26);(6,10)(7,11)(14,15)(18,19)(21,23)(22,26)(24,27)(25,28);(6,10)(8,12)(13,14,17,15)(16,19,20,18)(21,22,25,24)(23,27,28,26);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(13,15,17,14)(16,19,20,18)(21,24,25,22)(23,27,28,26);(5,8)(6,11)(7,10)(9,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331963, #72331994, #72332038
maximal normal subgroup of	#72331963, #72331994, #72332038
maximal quotient of	#72332619

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 898789);