#25194

order	256
nilpotency class	3
isoclinism	isoclinic class of #24292
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
prime radical	#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

490 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:

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

Polycyclic presentation:

2^8|a^2=f,b^2=f,c^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,d^b=dh,e^b=eh,f^c=fh,g^c=gh,e^d=eh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#25194, #6439024, #6439026, …
maximal normal subgroup of	C₂×#25194, #6439024, #6439026, …
maximal quotient of	#283711, #283713, #283719, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18182);