#996707

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

3190 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)(6,7)(8,12)(14,17)(15,19)(21,22,26,24)(23,28,27,25);(5,6,9,7)(8,10,12,11)(13,14,16,17)(15,18,19,20)(21,23,26,27)(22,25,24,28);(8,12)(10,11)(13,15,16,19)(14,18,17,20)(21,24,26,22)(23,25,27,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,18)(16,17)(19,20);(5,8)(6,10)(7,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,25)(22,27)(23,24)(26,28)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,d^2=fgh,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	#72331962, #72331971, #72332058
maximal normal subgroup of	#72331962, #72331971, #72332058
maximal quotient of	#72333162

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 903903);