#25189

order	256
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,11,15,12,16,10,26)(7,17,19,13,8,20,22,29)(9,23,24,14,25,30,27,31)(18,28,34,35,21,36,33,32);(1,2)(3,4)(5,7,11,19,12,8,10,22)(6,13,15,20,16,29,26,17)(9,18,24,34,25,21,27,33)(14,28,30,35,31,36,23,32);(2,4)(5,8)(6,13)(7,12)(9,21)(10,22)(11,19)(14,28)(15,17)(16,29)(18,25)(20,26)(23,35)(24,34)(27,33)(30,32)(31,36);(5,9,12,25)(6,14,16,31)(7,18,8,21)(10,27,11,24)(13,28,29,36)(15,30,26,23)(17,32,20,35)(19,34,22,33)

Polycyclic presentation:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#25189, #6439019, #6439026, …
maximal normal subgroup of	C₂×#25189, #6439019, #6439026, …
maximal quotient of	#283580, #283582, #283588, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18177);