#41842

order	256
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	Aut(G) = C₂¹⁵ (order 32768) · Inn(G) = C₂⁵ · Out(G) = C₂¹⁰
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

761 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10,13,16)(11,14,12,15)(17,18)(19,25)(20,26)(21,24)(22,23)(27,31)(28,32)(29,30);(1,2,4,6)(3,5,8,7)(9,11,13,12)(10,14,16,15)(18,23)(19,26)(24,30)(27,32);(3,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,32)(30,31);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,16,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32);(1,2)(3,5)(4,6)(7,8)(9,12,13,11)(10,15,16,14)(17,21,22,29)(18,24,23,30)(19,27,26,32)(20,28,25,31)

Polycyclic presentation:

2^8|a^2=gh,b^2=fg,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#41842, #7639826, #7640248, …
maximal normal subgroup of	C₂×#41842, #7639826, #7640248, …
maximal quotient of	#544911, #545310, #545967, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34830);