#23695

order	256
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	Aut(G) = C₂¹¹×D₄ (order 16384) · Inn(G) = C₂²×D₄ · Out(G) = C₂⁹
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂²×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

422 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=h,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,g^c=gh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#23695, #6432265, #6432276, …
maximal normal subgroup of	C₂×#23695, #6432265, #6432276, …
maximal quotient of	#243612, #243613, #243614, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16683);