#849923

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

3640 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)(9,11)(10,12)(14,17)(15,19);(5,7,9,12)(6,8,11,10)(13,14,16,17)(15,18,19,20)(21,22,24,26)(23,25,27,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20);(5,8,9,10)(6,7,11,12)(13,15)(14,18)(16,19)(17,20)(21,23,24,27)(22,25,26,28)

Polycyclic presentation:

2^9|b^2=fh,e^2=h,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	#72326510, #72326511, #72326543
maximal normal subgroup of	#72326510, #72326511, #72326543

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 757119);