#989141

order	512
nilpotency class	2
isoclinism	isoclinic class of #988785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2404	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3848 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)(7,8)(9,10)(11,12)(13,14,17,16)(15,19,20,18)(21,22,24,25)(23,27,28,26);(6,10)(7,11)(13,15)(14,18)(16,19)(17,20)(21,22)(23,27)(24,25)(26,28);(6,10)(8,12)(15,20)(18,19)(22,25)(26,27);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(13,16,17,14)(15,19,20,18)(21,23,24,28)(22,26,25,27);(5,8)(6,11)(7,10)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)

Polycyclic presentation:

2^9|a^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	#72232483, #72232499, #72232526
maximal normal subgroup of	#72232483, #72232499, #72232526
maximal quotient of	#72330685

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 896337);