#615581

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4278 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)(14,17)(15,19)(21,22,25,28)(23,27,24,26);(5,7,9,8)(6,10,12,11)(13,14,16,17)(15,18,19,20)(21,23,25,24)(22,26,28,27);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,27)(25,28);(5,6)(7,10)(8,11)(9,12);(5,8,9,7)(6,11,12,10)(13,15,16,19)(14,18,17,20)(21,24,25,23)(22,27,28,26)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264221, #72264232, #72264260, …
maximal normal subgroup of	#72264221, #72264232, #72264260, …
maximal quotient of	#72307083

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 522777);