#892354

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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

3638 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72326523, #72326528, #72326543
maximal normal subgroup of	#72326523, #72326528, #72326543

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 799550);