#851187

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2847	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3828 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,c^2=f,e^2=f,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	#72335368, #72335391, #72335414, …
maximal normal subgroup of	#72335368, #72335391, #72335414, …
maximal quotient of	#72335945, #72335950

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 758383);