#25145

order	256
nilpotency class	3
isoclinism	isoclinic class of #25140
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	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#3005	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

498 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

43 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#25145, #6438879, #6438882, …
maximal normal subgroup of	C₂×#25145, #6438879, #6438882, …
maximal quotient of	#281564, #281565, #281576, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18133);