#17328

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) = C₂²×#120×#392 (order 8192) · Inn(G) = C₂³ · Out(G) = C₂⁴×#392
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂³
commutator, socle	C₂²	C₂²×C₄²
Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

375 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=e,b^2=f,e^2=g,f^2=h,c^a=ch,d^c=dg

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#128434, #128435, #128436, …
maximal normal subgroup of	#128434, #128435, #128436, …
maximal quotient of	#180806, #180825, #180853, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10316);